Nerve injury when my nose was cauterized. is this permanent?

Unable to tell: With this information. It depends on how much cauterization, which nerves, etc. As a general rule, not specific to this, smaller nerves that are mildly damaged in minor surgeries often recover but will take sometimes six to nine months as nerves grow extremely slow compared to other tissue. While i can not be sure in your case with limited info, give it a year until you feel it is permanent.